\newproblem{lay:6_2_29}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 6.2.29}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $U$ and $V$ be $n\times n$ orthogonal matrices. Explain why $UV$ is an orthogonal matrix.
}{
   % Solution
	Let's calculate $(UV)^{-1}$
	\begin{center}
		$\begin{array}{rcll}
			(UV)^{-1}&=&V^{-1}U^{-1} & \text{Properties of matrix inverse; U,V are invertible} \\
			   &=&V^TU^T & \text{U and V are orthogonal matrices} \\
				 &=&(UV)^T & \text{Properties of matrix transpose} \\
		\end{array}$
	\end{center}
	So, $UV$ is invertible and its inverse is $(UV)^T$.
}
\useproblem{lay:6_2_29}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
